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The generic symplectic $$C^1$$-diffeomorphisms of four-dimensional symplectic manifolds are hyperbolic, partially hyperbolic or have a completely elliptic periodic point. (English) Zbl 1030.37037
The title of this paper reflects well the main result. More precisely, a refinement of a result of S. E. Newhouse [Am. J. Math. 99, 1061-1087 (1977; Zbl 0379.58011)] is given in a four-dimensional case.

##### MSC:
 37J10 Symplectic mappings, fixed points (dynamical systems) (MSC2010) 37C05 Dynamical systems involving smooth mappings and diffeomorphisms 37C20 Generic properties, structural stability of dynamical systems 37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) 37D30 Partially hyperbolic systems and dominated splittings 37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory, local dynamics
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